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11 Conclusions and Recommendations
Pages 407-432

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From page 407...
... Widespread failure to learn mathematics limits individual possibilities and hampers national growth. Our experiences, discussions, and review of the literature have convinced us that school mathematics demands substantial change.
From page 408...
... Then we propose changes needed in the curriculum if students are to develop mathematical proficiency, and we offer some recommendations for instruction. Finally, we discuss teacher preparation and professional development related to mathematics teaching, setting out recommendations designed to help teachers be more proficient in their work.
From page 409...
... Proficiency for all demands that fundamental changes be made concurrently in curriculum, instructional materials, classroom practice, teacher preparation, and professional development. These changes will require continuing, coordinated action on the part of policy makers, teacher educators, teachers, and parents.
From page 410...
... The balanced and integrated development of all five strands of mathematical proficiency requires that various elements of the school curriculumgoals, core content, learning activities, and assessment efforts be coordinated toward the same end. Achieving that coordination puts heavy demands on instructional programs, on the materials used in instruction, and on the way in which instructional time is managed.
From page 411...
... Parents and other caregivers, through games, puzzles, and other activities in the home, can also help children develop their informal knowledge and can augment the school's efforts. lust as adults in the home can help children avoid reading difficulties through activities that promote language and literacy growth, so too can they help children avoid difficulties in mathematics by helping them develop their informal knowledge of number, pattern, shape, and space.
From page 412...
... Creating this kind of learning will require changes in all parts of school mathematics to ensure that the following recommendations are implemented: · An integrated approach show/ loe taken to the deve/totn? ent of a/Y five strands of proficiency with who/se nun?
From page 413...
... Instructiona/t n? ateria/[s shou/(d inc/~ude visua/t and /linguistic supports to help students develop this representations/ aloi/tity.
From page 414...
... But the technology has not made obsolete the need to understand and be able to perform basic written algorithms for addition, subtraction, multiplication, and division of numbers, whether expressed as whole numbers, fractions, or decimals. Beyond providing tools for computation, algorithms can be analyzed and compared, which can help students understand the nature and properties of operations and of place-value notation for numbers.
From page 415...
... If children are not encouraged to use the mental computational procedures they have when entering school, those procedures will erode. But when instruction emphasizes estimation and mental arithmetic, conceptual understanding and fluency with mental procedures can be enhanced.
From page 416...
... Instructional sequences in which more time is spent at the outset on developing meaning for the various representations of rational numbers and the concept of unit have been shown to promote mathematical proficiency. Research reveals that the kinds of errors students make when beginning to operate with rational numbers often come because they have not yet developed meaning for these numbers and are applying poorly understood rules for whole numbers.
From page 417...
... Many school mathematics programs fail to develop children's understanding of ratio comparisons and move directly to formal procedures for solving missing-value proportion problems. Research tracing the development of proportional reasoning shows that proficiency grows as students develop and connect different aspects of proportional reasoning.
From page 418...
... The number line may become particularly useful as students are learning about integers and rational numbers, for it may help students develop a sense of the magnitudes and relationships of those numbers in a way that is less clear in other .
From page 419...
... New lines of research and development are focusing on ways that the elementary and middle school curriculum can be used to support the development of algebraic reasoning. These efforts attempt to avoid the difficulties many students now experience and to lay a better foundation for secondary school mathematics.
From page 420...
... We believe such efforts are virtually guarschool anteed to result in many students failing to develop proficiency in algebra, in students part because the transition to algebra is so abrupt. Instead, a different curriculum is needed for algebra in middle school: .
From page 421...
... Further, instructional materials in pre-K to grade 8 mathematics seldom provide the guidance and assistance that teachers in other countries find helpful, such as discussions of children's typical misconceptions or alternative solution methods. How teachers might understand and use instructional materials to help students develop mathematical proficiency is not well understood.
From page 422...
... Mathematical proficiency as we have defined it cannot be developed unless regular time (say, one hour each school day) is allocated to and used for mathematics instruction in every grade of elementary and middle school.
From page 423...
... The following recommendations will help make assessment more effective in developing mathematical proficiency: . Assessment, whether interns/ or e~cterna/t, show/ loe focused on the deve/totn?
From page 424...
... We believe that planning needs to reflect a deep and thorough consideration of the mathematical content of a lesson and of students' thinking and learning. Instructional materials need to support teachers in their planning, and teachers need to have time to plan.
From page 425...
... Managing Classroom Discourse Mathematics classrooms are more likely to be places in which mathematical proficiency develops when they are communities of learners and not collections of isolated individuals. Research on creating classrooms that function as communities of learners has identified several important features of these classrooms: ideas and methods are valued, students have autonomy in choosing and sharing solution methods, mistakes are valued as sites of learning for everyone, and the authority for correctness lies in logic and the structure of the subject, not in the teacher.
From page 426...
... Research reveals that various kinds of physical materials commonly used to help children learn mathematics are often no more concrete to them than symbols on paper might be. Concrete is not the same as physical.
From page 427...
... For example, students who use calculators tend to show improved conceptual understanding, greater ability to choose the correct operation, and greater skill in estimation and mental arithmetic without a loss of basic computational skills. They are also familiar with a wider range of numbers than students who do not use calculators and are better able to tackle realistic mathematics problems.
From page 428...
... The evidence indicates that these forms of knowledge are not acquired in conventional undergraduate mathematics courses, whether they are general survey courses or specialized courses for mathematics majors. The implications for teacher preparation and professional development are that teachers need to learn these forms of knowledge in ways that help them forge connections.
From page 429...
... The following recommendations reflect our judgment concerning the specialized knowledge that teachers need: Teachers of grades pre-K-8 show/ have a deep understanding of the n? athen?
From page 430...
... , its affiliated groups, or other organizations. These occasions can provide opportunities for professional development of the sort discussed above.
From page 431...
... Furthermore, these activities are often conducted by an array of professional developers with minimal qualifications in mathematics and mathematics teaching. Professional development in mathematics needs to be sustained over time that is measured in years, not weeks or months, and it needs to involve a substantial amount of time each year.
From page 432...
... Adopting mathematics textbooks from other countries, testing teachers, holding students back a grade, putting schools under state sanctions none of these alone will advance school mathematics very far toward mathematical proficiency for all. Instead, coordinated, systematic, and sustained modifications will need to be made in how school mathematics instruction has commonly proceeded, and support of new and different kinds will be required.


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